It is wrong to think the centers of compressions / rarefactions
of a longitudinal traveling wave are the counterparts of the
crests/ troughs of a transverse traveling wave. They are not because the displacements of the particles at the former
are zero while those at the latter displace from thier equilibrium positions the most.

Besides, particles at centers
of compressions (rarefactions) have the greatest speed, and moving in
the same (opposite) direction of the wave propagation. Particles at the
crests / troughs of a transverse traveling wave are momentarily at rest.

Actually, the compressions / rarefactions of a longitudinal wave correspond to the two "steepest hillsides" of a transerse wave.

In the figure below, the direction to the right (left) in the longitudinal wave is transformed to upward (downward) direction in the tansverse wave.

The author (Chiu-king Ng) has the copyright on all the simulations in this website.